= Logarithmic cancellation on an exterior annulus
{title2=$f(z)=2\log(1+1/z)-2\log(1-1/z)$}
For $|z|>1$, both logarithms in the display can use their <principal logarithms>. The <real part> is $\log(|z+1|^2/|z-1|^2)$ and the <derivative> is $2/(z+1)-2/(z-1)$. The two <residues> cancel on a loop surrounding both points. Equivalently $f=4\sum_{m\ge0}z^{-(2m+1)}/(2m+1)$ is globally single-valued outside the unit <circle>, although individual logarithms of $z+1$ and $z-1$ are not.
// Destination: complex-dynamics.bigb
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